Optimal. Leaf size=65 \[ -\frac{d (b c-a d)}{3 b^3 (a+b x)^6}-\frac{(b c-a d)^2}{7 b^3 (a+b x)^7}-\frac{d^2}{5 b^3 (a+b x)^5} \]
[Out]
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Rubi [A] time = 0.0993778, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ -\frac{d (b c-a d)}{3 b^3 (a+b x)^6}-\frac{(b c-a d)^2}{7 b^3 (a+b x)^7}-\frac{d^2}{5 b^3 (a+b x)^5} \]
Antiderivative was successfully verified.
[In] Int[(a*c + (b*c + a*d)*x + b*d*x^2)^2/(a + b*x)^10,x]
[Out]
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Rubi in Sympy [A] time = 24.2879, size = 54, normalized size = 0.83 \[ - \frac{d^{2}}{5 b^{3} \left (a + b x\right )^{5}} + \frac{d \left (a d - b c\right )}{3 b^{3} \left (a + b x\right )^{6}} - \frac{\left (a d - b c\right )^{2}}{7 b^{3} \left (a + b x\right )^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a*c+(a*d+b*c)*x+b*d*x**2)**2/(b*x+a)**10,x)
[Out]
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Mathematica [A] time = 0.0472695, size = 57, normalized size = 0.88 \[ -\frac{a^2 d^2+a b d (5 c+7 d x)+b^2 \left (15 c^2+35 c d x+21 d^2 x^2\right )}{105 b^3 (a+b x)^7} \]
Antiderivative was successfully verified.
[In] Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)^2/(a + b*x)^10,x]
[Out]
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Maple [A] time = 0.009, size = 71, normalized size = 1.1 \[ -{\frac{{d}^{2}}{5\,{b}^{3} \left ( bx+a \right ) ^{5}}}+{\frac{ \left ( ad-bc \right ) d}{3\,{b}^{3} \left ( bx+a \right ) ^{6}}}-{\frac{{a}^{2}{d}^{2}-2\,cabd+{b}^{2}{c}^{2}}{7\,{b}^{3} \left ( bx+a \right ) ^{7}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a*c+(a*d+b*c)*x+x^2*b*d)^2/(b*x+a)^10,x)
[Out]
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Maxima [A] time = 0.760787, size = 177, normalized size = 2.72 \[ -\frac{21 \, b^{2} d^{2} x^{2} + 15 \, b^{2} c^{2} + 5 \, a b c d + a^{2} d^{2} + 7 \,{\left (5 \, b^{2} c d + a b d^{2}\right )} x}{105 \,{\left (b^{10} x^{7} + 7 \, a b^{9} x^{6} + 21 \, a^{2} b^{8} x^{5} + 35 \, a^{3} b^{7} x^{4} + 35 \, a^{4} b^{6} x^{3} + 21 \, a^{5} b^{5} x^{2} + 7 \, a^{6} b^{4} x + a^{7} b^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*d*x^2 + a*c + (b*c + a*d)*x)^2/(b*x + a)^10,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.201754, size = 177, normalized size = 2.72 \[ -\frac{21 \, b^{2} d^{2} x^{2} + 15 \, b^{2} c^{2} + 5 \, a b c d + a^{2} d^{2} + 7 \,{\left (5 \, b^{2} c d + a b d^{2}\right )} x}{105 \,{\left (b^{10} x^{7} + 7 \, a b^{9} x^{6} + 21 \, a^{2} b^{8} x^{5} + 35 \, a^{3} b^{7} x^{4} + 35 \, a^{4} b^{6} x^{3} + 21 \, a^{5} b^{5} x^{2} + 7 \, a^{6} b^{4} x + a^{7} b^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*d*x^2 + a*c + (b*c + a*d)*x)^2/(b*x + a)^10,x, algorithm="fricas")
[Out]
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Sympy [A] time = 6.82502, size = 139, normalized size = 2.14 \[ - \frac{a^{2} d^{2} + 5 a b c d + 15 b^{2} c^{2} + 21 b^{2} d^{2} x^{2} + x \left (7 a b d^{2} + 35 b^{2} c d\right )}{105 a^{7} b^{3} + 735 a^{6} b^{4} x + 2205 a^{5} b^{5} x^{2} + 3675 a^{4} b^{6} x^{3} + 3675 a^{3} b^{7} x^{4} + 2205 a^{2} b^{8} x^{5} + 735 a b^{9} x^{6} + 105 b^{10} x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a*c+(a*d+b*c)*x+b*d*x**2)**2/(b*x+a)**10,x)
[Out]
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GIAC/XCAS [A] time = 0.20943, size = 82, normalized size = 1.26 \[ -\frac{21 \, b^{2} d^{2} x^{2} + 35 \, b^{2} c d x + 7 \, a b d^{2} x + 15 \, b^{2} c^{2} + 5 \, a b c d + a^{2} d^{2}}{105 \,{\left (b x + a\right )}^{7} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*d*x^2 + a*c + (b*c + a*d)*x)^2/(b*x + a)^10,x, algorithm="giac")
[Out]